Let us consider the binary classification setting. In the work of [1]}, it was shown that the (combinatorial version) of the Sauer-Shelah-Perles (SSP) lemma fails in the partial concepts setting. In the total concepts setting, this lemma controls the size of a concept class \(\mathcal {H}\)  in terms of its VC dimension. Another variant of this lemma controls the growth function of the class. Given a set \(C = \lbrace x_1,...,x_m\rbrace  \subseteq \mathcal {X}\) , the growth function of \(\mathcal {H}\subseteq \lbrace 0,1\rbrace ^\mathcal {X}\)  with respect to \(C\)  is the cardinality of the set \(\Pi _{\mathcal {H}}(C)\)  of binary patterns realized by hypotheses in \(\mathcal {H}\)  when projected to \(C\) , i.e.,
\(\Pi _\mathcal {H}(C) = \lbrace  (h(x_1),...,h(x_m)) : h \in \mathcal {H}\rbrace  \subseteq \lbrace 0,1\rbrace ^m\,.\) 
